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1500 questions
67
votes
8 answers
Example of Partial Order that's not a Total Order and why?
I'm looking for a simple example of a partial order which is not a total order so that I can grasp the concept and the difference between the two. An explanation of why the example is a partial order but not a total order would also be greatly…
eZanmoto
- 789
67
votes
5 answers
When is the closure of an intersection equal to the intersection of closures?
We know $\overline{\bigcap A_{\alpha}}\subseteq\bigcap\overline{A}_{\alpha} $, but when is the reverse inclusion true? Can you give some properties of the underlying space that would guarantee this?
Forever Mozart
- 9,594
67
votes
1 answer
Proving that $\int_0^\pi\frac{x\ln(1-\sin x)}{\sin x}dx=3\int_0^\frac{\pi}{2}\frac{x\ln(1-\sin x)}{\sin x}dx$
Prove without evaluating the integrals that: $$2\int_0^\frac{\pi}{2}\frac{x\ln(1-\sin x)}{\sin x}dx=\int_\frac{\pi}{2}^\pi\frac{x\ln(1-\sin x)}{\sin x}dx\label{*}\tag{*}$$ Or equivalently: $$\boxed{\int_0^\pi\frac{x\ln(1-\sin x)}{\sin…
Zacky
- 30,834
67
votes
4 answers
Given a Fibonacci number , find the next Fibonacci number
The Fibonacci sequence is $0, 1, 1, 2, 3, 5, 8, 13, 21, 34,\ldots$ , where each term after the first two is the sum of the two previous terms. Can we find the next Fibonacci number if we are given any Fibonacci number? For example, if $n = 8$ then…
ppSpp
- 951
67
votes
7 answers
Could I be using proof by contradiction too much?
Lately, I've developed a habit of proving almost everything by contradiction. Even for theorems for which direct proofs are the clear choice, I'd just start by writing "Assume not" then prove it directly, thereby reaching a "contradiction." Is this…
user64844
- 653
67
votes
1 answer
Abstract nonsense proof of snake lemma
During my studies, I always wanted to see a "purely category-theoretical" proof of the Snake Lemma, i.e. a proof that constructs all morphisms (including the snake) and proves exactness via universal properties. It was an interest little shared by…
Jesko Hüttenhain
- 15,286
67
votes
3 answers
Why is the localization at a prime ideal a local ring?
I would like to know, why $ \mathfrak{p} A_{\mathfrak{p}} $ is the maximal ideal of the local ring $ A_{\mathfrak{p}} $, where $ \mathfrak{p} $ is a prime ideal of $ A $ and $ A_{\mathfrak{p}} $ is the localization of the ring $ A $ with respect to…
Bryan
- 885
67
votes
8 answers
How much Math do you REALLY do in your job?
I am writing this, as I am a currently an intern at an aircraft manufactur. I am studying a mixture of engineering and applied math. During the semester I focussed on numerical courses and my applied field is CFD. Even though every mathematician…
Thomas
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67
votes
5 answers
Is there any branch of Mathematics which has no applications in any other field or in real world?
Is there any branch of Mathematics which has no applications in any other field or in real world ? for instance , maybe : number theory ? mathematical logic ? is there something like this ?
FNH
- 9,552
- 4
- 67
- 118
67
votes
6 answers
What is the difference between homotopy and homeomorphism?
What is the difference between homotopy and homeomorphism? Let X and Y be two spaces, Supposed X and Y are homotopy equivalent and have the same dimension, can it be proved that they are homeomorphic? Otherwise, is there any counterexample?…
liufu
- 731
67
votes
4 answers
Counterexample Math Books
I have been able to find several counterexample books in some math areas. For example: $\bullet$ Counterexamples in Analysis, Bernard R. Gelbaum, John M. H. Olmsted $\bullet$ Counterexamples in Topology, Lynn Arthur Steen, J. Arthur Seebach Jr.…
Amzoti
- 56,689
67
votes
8 answers
How does a non-mathematician go about publishing a proof in a way that ensures it to be up to the mathematical community's standards?
I'm a computer science student who is a maths hobbyist. I'm convinced that I've proven a major conjecture. The problem lies in that I've never published anything before and am not a mathematician by profession. Knowing full well that my proof may be…
Balbanna
- 713
67
votes
4 answers
Calculate on which side of a straight line is a given point located?
I am a programmer without really good knowledge in math. :/ So I have to write an algorithm that changes the color of pixel(dot) P to opposite if it's on left side of the straigt line in coordinate system (and the line is not vertical, with that I…
Ritvars
- 773
67
votes
9 answers
Prove every odd integer is the difference of two squares
I know that I should use the definition of an odd integer ($2k+1$), but that's about it. Thanks in advance!
papercuts
- 1,893
67
votes
16 answers
How can I introduce complex numbers to precalculus students?
I teach a precalculus course almost every semester, and over these semesters I've found various things that work quite well. For example, when talking about polynomials and rational functions, in particular "zeroes" and "vertical asymptotes", I…
user641